Finite Group Databasebeta

← #27156855

Subgroups — #27156855

Order 12416.

122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261639C211yesmaximal, characteristic, complemented
2#20261593C211yesmaximal, characteristic, complemented
3#20260903C211yesmaximal, characteristic
4#15690877—21no—
5#15690869—21no—
6#15690835—21no—
7#15690806C2211yescharacteristic
8#15690737—21no—
9#15690727C411yescharacteristic, complemented
10H(5,𝔽2)×C97C411yescharacteristic, complemented
11C2×#11715897—41no—
12C22⋊#87604—41no—
13C2×D388D411yescharacteristic
14C2×C22⋊D97—41no—
15#13095498—41no—
16#13095496—41no—
17#13095495—21no—
18C4×Dic97D411yescharacteristic
19#41×C97—21no—
20#41×C97—41no—
21C2×D4×C97—41no—
22C2×D4×C97—41no—
23C2×D4×C97C2×C411yescharacteristic
24C23×C97—21no—
25C23×C97—41no—
26C22×D97—21no—
27C2×#87604—81no—
28Q8×C97—21no—
29D4×C97—21no—
30D4×C97—41no—
31D4×C97—42no—
32D4×C97—41no—
33C2×C4×C97—41no—
34C2×C4×C97—41no—
35C2×C4×C97C22⋊C411yescharacteristic
36C22⋊D97—81no—
37C22⋊D97—81no—
38C2×Dic97—21no—
39D388—41no—
40#11715897—81no—
41#11715895—41no—
42C22×C97—41no—
43C22×C97—41no—
44C22×C97—81no—
45C22×C97—161no—
46C22×C97#9911yescharacteristic
47C2×D97—41no—
48C2×D97—81no—
49#87604—161no—
50C4×C97—21no—
51C4×C97—41no—
52Dic97—21no—
53Dic97—21no—
54C2×C97—21no—
55C2×C97—81no—
56C2×C97—81no—
57C2×C97AΣL(1,𝔽16)11yescharacteristic
58D97—81no—
59#1402—971nomaximal, Sylow
60C97#140211yescharacteristic, complemented, Sylow
61#453—971no—
62#353—971no—
63#331—971no—
64H(5,𝔽2)#8760411yescharacteristic, complemented
65AGL(1,ℤ/8)—1941no—
66#127—971no—
67C4⋊C8—1941no—
68#104—1941no—
69#100—971no—
70#99—1941no—
71#41—21no—
72#41—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—971no—
76C2×D4—3881no—
77C2×D4C2×#8760411yescharacteristic
78SD16—3881no—
79D8—3881no—
80M16—1941no—
81C2×C8—3881no—
82C22⋊C4—3881no—
83C42—971no—
84C23—21no—
85C23—41no—
86C23—1941no—
87Q8—21no—
88D4—21no—
89D4—41no—
90D4—42no—
91D4—41no—
92D4—7761no—
93D4—7761no—
94D4—3881no—
95C2×C4—41no—
96C2×C4—41no—
97C2×C4—7761no—
98C2×C4—1941no—
99C2×C4C22⋊#8760411yescharacteristic
100C8—7761no—
101C8—3881no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—7761no—
106C22—161no—
107C22—3881no—
108C22#1569086811yescharacteristic
109C4—21no—
110C4—41no—
111C4—15521no—
112C4—1941no—
113C4—1941no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—7761no—
118C2#2026163511yescharacteristic
subgroupG/H
centerC2#20261635
commutatorC2×D4×C97C2×C4
FrattiniC2×D4C2×#87604
FittingH(5,𝔽2)×C97C4
socleC2×C97AΣL(1,𝔽16)
prime radical#20260903C2
cosocle#15690806C22
hypercenterH(5,𝔽2)#87604
hypocenterC97#1402
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20261639C2semidirect
2. #20261593C2semidirect
9. #15690727C4semidirect
10. H(5,𝔽2)×C97C4semidirect
60. C97#1402semidirect
64. H(5,𝔽2)#87604semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202616392
#202615932
#202609032
#140297